Geometric Sequences

A geometric sequence is a sequence of numbers in which the ratio between consecutive terms is constant.

We can write a formula for the n th term of a geometric sequence in the form

a n =a r n ,

where r is the common ratio between successive terms.

Example 1:

{ 2,6,18,54,162,486,1458,... }

is a geometric sequence where each term is 3 times the previous term.

A formula for the n th term of the sequence is

a n = 2 3 ( 3 ) n

Example 2:

{ 12,6,3, 3 2 , 3 4 , 3 8 , 3 16 ,... }

is a geometric series where each term is 1 2 times the previous term.

A formula for the n th term of this sequence is

a n =24 ( 1 2 ) n

Example 3:

{ 1,2,6,24,120,720,5040,... }

is not a geometric sequence. The first ratio is 2 1 =2 , but the second ratio is 6 2 =3 .

No formula of the form

a n =a r n can be written for this sequence.

See also arithmetic sequences.